A362174 Number of n X n matrices with nonnegative integer entries such that the sum of the elements of each row is equal to the index of that row.
1, 1, 6, 180, 28000, 23152500, 103507455744, 2532712433771520, 342315030877028352000, 257389071045194840814562500, 1082814493908215083601185600000000, 25605944807023092680403880661295843852288, 3416912747607221845915134383073991514372073062400
Offset: 0
Keywords
Examples
a(1) = 1 as the only 1 X 1 matrix that satisfies the constraints is [1]. a(2) = 6 as there are 2 2d-vectors within the constraints with components that sum to 1 and independently 3 2d-vectors within the constraints with components that sum to 2. They are as follows: [[0 1],[1 1]], [[0 1],[2 0]], [[0 1],[0 2]], [[1 0],[1 1]], [[1 0],[2 0]], [[1 0],[0 2]], a(3) = 180 as there are 3 3d-vectors within the constraints with components that sum to 1, 6 that sum to 2, and 10 that sum to 3. 3*6*10 = 180.
Links
- Michael Richard, Table of n, a(n) for n = 0..52
Programs
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Maple
a:= n-> mul(binomial(n+k-1, n-1), k=1..n): seq(a(n), n=0..15);
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Mathematica
a[n_] := Product[Binomial[n + k - 1, n - 1], {k, 1, n}] -
PARI
a(n) = prod(k=1, n, binomial(n+k-1,n-1)); \\ Michel Marcus, Jun 25 2023
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Python
from math import comb, prod def a(n): return prod(comb(n+k, n-1) for k in range(n))
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Python
from math import factorial from functools import lru_cache @lru_cache(maxsize=None) def A362174(n): return A362174(n-1)*(2*n-1)*factorial(2*n-2)**2//n//factorial(n-1)**3//(n-1)**(n-1) if n else 1 # Chai Wah Wu, Jun 26 2023
Formula
a(n) = Product_{k=1..n} binomial(n+k-1,n-1).
a(n) = a(n-1)*(2n-1)*(2n-2)!^2/(n*(n-1)!^3*(n-1)^(n-1)). - Chai Wah Wu, Jun 26 2023
a(x) = x^x*G(2x+1)*(G(x+1)^(x-1)/G(x+2)^(x+1)) where G(x) is the Barnes G-function is a differentiable continuation of a(n) to the nonnegative reals. - Michael Richard, Jun 27 2023
a(n) ~ A * 2^(2*n^2 - n/2 - 7/12) / (Pi^((n+1)/2) * exp(n^2/2 - n + 1/6) * n^(n/2 + 5/12)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 19 2023
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